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Jack Good was always willing to work things out in his own way. His personal resume describes how he independently rediscovered the irrationality of the square root of two at the age of nine, the principle of mathematical induction at the age of 13, and Fermat's theorem on sums of two squares as a sixth former. His capacity for independent, original thought saw him recognised as a mathematical prodigy and he studied at Cambridge University, completing a doctorate on Fourier analysis under the great pure mathematician G. H. Hardy in 1941.

It was both his aptitude for mathematics and for chess1 that brought him to the attention of Hugh Alexander and Bletchley Park. He was initially posted to Hut 8 which provided a whole new outlet for his ingenuity, working German naval Enigma traffic. Good's cryptanalytic career got off to a rocky start when he annoyed the head of Hut 8 (one Alan Turing) by napping on a night shift2. For a while Turing refused to acknowledge Good, but soon Good was able to prove his worth and one of Bletchley's most successful partnerships took off.

Turing and Good seem to be kindred spirits and would discuss many topics of mutual interest such as mathematics, computation, artificial intelligence, and philosophy; they would play Go3 at Turing's billet in The Crown Inn. Most importantly, they would discuss statistics; a topic which neither of the two pure mathematicians had devoted much time to at Cambridge, but which they now realised was a critical tool in cryptanalysis. Between them they set about developing their own statistical ideas, some of which were rediscoveries and some of which were new. This could be viewed as the dawn of a new age in statistics as they promoted a viewpoint now known as Bayesian statistics4. It was a viewpoint that Good championed to great effect for the rest of his life.

Statistics begins by thinking about a fixed underlying truth and reasons about what events one might observe, but the reasoning can also be reversed and given an observed set of events one can try to reason about the underlying truth. Classical statistics treats the underlying truth as fixed and immutable, but unknown; Bayesian statistics treats the underlying truth as a competing set of explanations with associated level of belief. Good and Turing found that cryptanalysis acquires a much stronger expression in Bayesian statistics. They were particularly interested in cases where observations arrived sequentially over time and how the Bayesian beliefs could be updated continually. This led to results such as Good-Turing frequency estimation which allows measurement of the probability of observing an event which had not occurred in previous observations. Ideas such as these had a great impact on the analysis of the naval Enigma via processes such as Banburismus scoring and also on Good's later work under Max Newman attacking the Fish ciphers of the German High Command.

Equally important to Bletchley was Good's intuition for computation and how to use the new machines such as the Newmanry's COLOSSUS to best effect. In one throwaway anecdote, Good tells how he was able to double the efficiency of the COLOSSUS simply by reducing the precision to which one of the statistical scores were computed (using his statistical expertise to show that the error introduced was not great enough to change the conclusions).

After the war, Max Newman quickly recruited Good as a lecturer at Manchester University with the intent of also using him to help develop the Manchester Mark I computer. A dislike of lecturing and frustrations at finding a publisher for his (now classic) treatise on Bayesian statistics "Probability and the Weighing of Evidence" drew Good away from academia and returned him to classified work at GCHQ. The Cold War and the growth of computational power meant Good's talents were incredibly valuable to GCHQ as he continued to contribute to the revolution in cryptanalysis brought about by efficient computers. It was during this time that Good developed a powerful computational technique which is the basis of modern methods known as Fast Fourier Transforms (FFTs). FFTs have found an immense range of applications from signal processing, to solving differential equations, to providing the best known methods for a computer to multiply large numbers. They massively reduce the computational power required to achieve these tasks and the idea of the FFT has been described as the most important algorithm in modern computation. Despite his continued success, Good was still unsettled. He nominally left GCHQ after about a decade while maintaining some links and then flitted between academia, government, and the computer industry for several years.

Taking his leave from the classified world freed up more time for Good to spend writing and his (already) healthy rate of publication accelerated. A prolific, engaging and clear writer, Good was willing to pontificate on a wide range of topics from the deeply technical to the whimsical5. His list of publications runs to over 2,000 items and over 3,000,000 words and it is impossible to talk in any detail about its full breadth. One article did lead to another interesting side project. In his 1965 paper "Speculations concerning the first ultraintelligent machine", Good remarked that if humans can ever design an artificial intellect superior to their own then that intellect could presumably build an even more powerful artificial intellect and so on recursively. This would lead to an exponential growth in artificial intelligence that is now termed the "technological singularity". The exponential model is not unreasonable; a similar phenomenon has been seen with Moore's Law where improved design of microcircuitry leads to yet more improvements in the design of microcircuitry.

What was probably an amusing intellectual exercise for Good drew the attention of many people with a passing interest in artificial intelligence. Among them was Stanley Kubrick who was interested in these themes for his film "2001: A Space Odyssey"; Kubrick engaged Good as a technical consultant to advise on the character of the HAL 9000 supercomputer. Good's connections with the film industry must have continued, for in 1995 he was elected a member of the Academy of Motion Pictures Arts and Sciences. This is probably a unique achievement amongst research mathematicians.

The word unique is probably the last word on Good's personality: a lively and ecumenical intellect of infectious enthusiasm and mischievous wit. This brief article is not enough to do him full justice and the interested reader is encouraged to find out more about Good in his own words6 either via an extended interview or his idiosyncratically titled "Good Thinking: The Foundations of Probability and Its Applications".

A GCHQ senior mathematician

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Footnotes

1. Good claims to have competed on even terms with chess champions and fellow Bletchley Park codebreakers Hugh Alexander and Harry Golombek, and with women’s world champion Vera Menchik at speed chess, though not at "serious" chess.

2. A subsequent tale tells how Good gained an insight into breaking a particular Enigma message in his sleep and so perhaps his napping can be forgiven.

3. An Asian board game. Good was able to offer Turing a significant handicap of six stones.

4. The term "Bayesian" in the English language was first used by Good, as was the important concept of a "Bayes factor".

5. His papers "A classification of rules for writing informative English", "A classification of fallacious arguments and interpretations" and "How rational should a manager be?" are all recommended.

6. Good would regularly style himself as the foremost expert on the life and works of Jack Good.